The SSM Procedure (Experimental)

TREND Statement

TREND Statement Options

TREND name ( type ) <options> ;

The TREND statement defines a trend term in the model. Loosely speaking, a trend is a special type of component that captures the time-varying level of the data. The options in the TREND statement enable you to specify a wide variety of commonly used trend patterns. Each TREND specification in effect stands for a special pair of STATE and COMPONENT statements. You can specify more than one TREND statement. Each separate trend specification defines a component that is assumed to be independent of all other component specifications in the model.

You can refer to the state associated with a TREND specification by appending the string “_state_”at the end of its name. For example, name_state_ is the state associated with a trend named name. You can use name_state_ in a COMPONENT statement to define a linear combination of its elements. The estimate of this linear combination can then be printed or output to a data set. The nominal dimension of name_state_ is taken to be 1, or the number of variables in the list that is specified in the CROSS= option in the TREND statement that is used to define name.

Some of these trend specifications are applicable to all the data types—that is, they can be used for both regular data types and irregular data types, while the others require that the data be regular or regular-with-replication. Of course, the trend specification is only a part of the overall model specification. Therefore, the other parts of the model can imply additional constraints on the data type.

Table 27.3 lists the available trend models and their data requirements. The type column shows the admissible keywords that signify the particular trend type. For brevity, the Data Type column in Table 27.3 groups the regular and regular-with-replication data types into one category: regular. The section Predefined Trend Models provides additional details about these trend models.

Table 27.3: Summary of Trend Types

type	Data Type	Description	Parameters
ARIMA(P=integer D=integer …)	Regular	ARIMA trend specification	AR and MA coefficients,
			and the error variance $\sigma ^2$
DLL	Regular	Damped local linear	Level and slope $\sigma ^{2}_{1}$ , $\sigma ^{2}_{2}$ ,
			damping factor $\phi$
LL	Regular	Local linear	Level and slope $\sigma ^{2}_{1}$ , $\sigma ^{2}_{2}$
RW	Regular	Random walk	Level $\sigma ^2$
DECAY	Irregular	A type of decay pattern	Level $\sigma ^2$ , decay rate $\phi$
DECAY(OU)	Irregular	Ornstein-Uhlenbeck decay pattern	Level $\sigma ^2$ , decay rate $\phi$
GROWTH	Irregular	A type of growth pattern	Level $\sigma ^2$ , growth rate $\phi$
GROWTH(OU)	Irregular	Ornstein-Uhlenbeck growth pattern	Level $\sigma ^2$ , growth rate $\phi$
PS(order)	Irregular	Polynomial spline of order up to 3	Level $\sigma ^2$

The keyword specification of different trend types, except possibly the ARIMA trend, is quite simple. For example, the following statement specifies polySpline as a trend of type polynomial spline of order 2:

      trend polySpline(ps(2));

Similarly, the following statement defines dampedTrend as a damped local linear trend:

      trend dampedTrend(dll) slopevar=x;

The variance parameter that governs the slope equation of this trend type is given by a variable x, which must be defined elsewhere in the program. The other parameters that define dampedTrend are left unspecified.

The ARIMA trend specification permits specification of trends that follow an ARIMA(p,d,q) ${\times }$ (P,D,Q) $_{\mi {s}}$ model. The specification of ARIMA models requires some notation, which is explained first.

Let denote the backshift operator—that is, for any sequence $\mu _ t$ , $B \mu _ t = \mu _{t-1}$ . The higher powers of represent larger shifts (for example, $B^3 \mu _ t = \mu _{t-3}$ ). A random sequence $\mu _ t$ follows an ARIMA(p,d,q) ${\times }$ (P,D,Q) $_{\mi {s}}$ model with nonseasonal autoregressive order , seasonal autoregressive order , nonseasonal differencing order , seasonal differencing order , nonseasonal moving average order , and seasonal moving average order if it satisfies the following difference equation that is specified in terms of the polynomials in the back-shift operator, where is a white noise sequence and is the season length:

$\phi (B) \Phi (B^ s) (1-B)^{d} (1 - B^{s})^{D} \mu _ t = \theta (B) \Theta (B^ s) a_ t$

The polynomials $\phi , \Phi , \theta ,$ and $\Theta$ are of orders , , , and , respectively, which can be any nonnegative integers. The season length must be a positive integer. For example, $\mu _ t$ satisfies an ARIMA(1,0,1) model (that is, and ) if

$\mu _ t = \phi _{1} \mu _{t-1} + a_ t - \theta _{1} a_{t-1}$

for some coefficients $\phi _{1}$ and $\theta _{1}$ and a white noise sequence . Similarly $\mu _ t$ satisfies an ARIMA(0,1,1) ${\times }$ (0,1,1) $_{12}$ model if

$\mu _ t = \mu _{t-1} + \mu _{t-12} - \mu _{t-13} + a_ t - \theta _{1} a_{t-1} - \Theta _{1} a_{t-12} + \theta _{1} \Theta _{1} a_{t-13}$

for some coefficients $\theta _{1}$ and $\Theta _{1}$ and a white noise sequence . An ARIMA process is mean-zero, stationary, and invertible if , and the defining polynomials $\phi , \Phi , \theta ,$ and $\Theta$ have all their roots outside the unit circle—that is, their absolute values are strictly larger than 1.0. It is assumed that the coefficients of the polynomials $\phi , \Phi , \theta ,$ and $\Theta$ are constrained so that the stationarity and invertibility conditions are satisfied. The unknown coefficients of these polynomials become part of the model parameter vector that is estimated by using the data. The general form of ARIMA trend specification is as follows:

ARIMA(<P=integer> <D=integer> <Q=integer> <SP=integer> <SD=integer> <SQ=integer> <S=integer> )

By default, the different orders are equal to 0 and the season length is equal to 1. The following examples illustrate a few different ARIMA trend specifications:

This statement defines ima as an integrated moving average trend:

      trend ima(arima(d=1 q=1));

This statement defines airTrend as a trend that satisfies the well-known Airline model (ARIMA(0,1,1)(0,1,1)12 model) for monthly seasonal data:

      trend airTrend(arima(d=1 q=1 sd=1 sq=1 s=12));

This statement defines arma11 as a zero-mean ARMA(1,1) trend with autoregressive parameter fixed to 0.1:

      trend arma11(arima(p=1 q=1)) ar=0.1;

For an example of the use of ARIMA trend specification, see the example Example 27.6.

TREND Statement Options

You can use the following options in the TREND statement to specify the trend parameters and to request printing of the trend estimates. In addition, you can create a custom combination of given trend type by specifying the CROSS= option to create a more general trend. For an example of the use of the CROSS= option, see the section Getting Started: SSM Procedure and the discussion of the second model in Example 27.4.

AR= $\phi _1$ $\phi _2$ … $\phi _ p$

lists the values of the coefficients of the nonseasonal autoregressive polynomial

$\phi (B) = 1 - \phi _1 B - \ldots - \phi _{p} B^ p$

where the order is specified in the ARIMA trend specification. The coefficients $\phi _ i$ must define a stationary autoregressive polynomial.

CROSS=(var1, var2, ...) CROSS(MATCHPARM)=(var1, var2, ...)

creates a linear combination of one or more independent trend components that is based on the variables in the list. If the parameters of the trend are specified by options such as the LEVELVAR= option or the PHI= option, these parameters are shared by these constituent trends. For example, suppose that the CROSS= list contains two variables $(X_{1}$ and $X_{2})$ and the trend specification is of the type RW. The effect of CROSS=( $X_{1}, X_{2}$ ) is to create a component $\mu _{t} = X_{1}\mu _{1,t} + X_{2}\mu _{2,t}$ , where $\mu _{1,t}$ and $\mu _{2,t}$ are two independent random walk trends. Moreover, if the random walk trend specification uses the LEVELVAR= option to specify the variance parameter, $\mu _{1,t}$ and $\mu _{2,t}$ share the same variance parameter; otherwise, two separate variance parameters are assigned to these random walks. If the second form of the CROSS option, CROSS(MATCHPARM)=, is used, then the constituent trends share all the relevant parameters no matter how they are specified. The CROSS= option is useful for a variety of situations. For example, suppose is an indicator variable that is 1 before a certain time point and 0 thereafter. Then CROSS=(X) has the effect of turning off the trend component after time . Similarly, suppose $G_{1}$ and $G_{2}$ are indicators for gender—for example, $G_{1}$ = (GENDER=“M”) and $G_{2}$ = (GENDER=“F”). Then CROSS=( $G_{1}, G_{2}$ ) creates a trend that varies with the gender of the observation. The variables in the CROSS= list must be free of unknown parameters.

The CROSS= option can be computationally expensive; computationally it is equivalent to specifying as many separate trends as the number of variables in the specified list.

LEVELVAR=variable | number

specifies the disturbance variance parameter for all the trend types. For trend types LL and DLL, this option specifies $\sigma ^{2}_{1}$ . Any nonnegative value, including 0, is permissible. If variable contains unknown parameters, they are estimated from the data. Similarly, if the LEVELVAR= option is not specified, $\sigma ^{2}$ is estimated from the data.

MA= $\theta _1$ $\theta _2$ … $\theta _ q$

lists the values of the coefficients of the nonseasonal moving average polynomial,

$\theta (B) = 1 - \theta _1 B - \ldots - \theta _{q} B^ q$

where the order is specified in the ARIMA trend specification. The coefficients $\theta _ i$ must define an invertible moving average polynomial.

NODIFFUSE

causes the diffuse elements in the initial state of the state subsection underlying the trend component to be treated as nondiffuse. This option is applicable to all trend types except ARIMA. For the ARIMA trend type, this option is ignored even if the nonseasonal or seasonal differencing orders are nonzero. The diffuse elements are assumed to be independent, zero-mean, Gaussian variables. Their variances become part of the parameter vector and are estimated by using the data. This option is useful for creating a trend component that can be interpreted as a deviation from an overall trend component (with diffuse initialization), which is defined separately.

PHI=variable | number

specifies the value of $\phi$ for trend types DLL, DECAY, DECAY(OU), GROWTH, and GROWTH(OU). For the type DLL, the specified value must be between 0.0 and 1.0. For types DECAY and DECAY(OU), $\phi$ must be strictly negative. For types GROWTH and GROWTH(OU), $\phi$ must be strictly positive. If variable contains unknown parameters, they are estimated from the data. Similarly, if the PHI= option is not specified, $\phi$ is estimated from the data.

PRINT=COV | COV1 | FILTER | SMOOTH | T PRINT=(<COV> <COV1> <FILTER> <SMOOTH> <T> )

requests printing of the respective system matrices of the state equation underlying the specified trend, and the printing of its filtered and smoothed estimates. If any of these matrices are time-varying, the matrix that corresponds to the first time instance is printed.

SAR= $\Phi _1$ $\Phi _2$ … $\Phi _ P$

lists the values of the coefficients of the seasonal autoregressive polynomial

$\Phi (B^ s) = 1 - \Phi _1 B^ s - \ldots - \Phi _{P} B^{s P}$

where the order is specified by using the SP= option in the ARIMA trend specification and the season length is specified in the S= option. The coefficients $\Phi _ i$ must define a stationary autoregressive polynomial.

SMA= $\Theta _1$ $\Theta _2$ … $\Theta _ Q$

lists the values of the coefficients of the seasonal moving average polynomial

$\Theta (B^ s) = 1 - \Theta _1 B^ s - \ldots - \Theta _{Q} B^{s Q}$

where the order is specified by using the SQ= option in the ARIMA trend specification and the season length is specified in the S= option. The coefficients $\Theta _ i$ must define an invertible moving average polynomial.

SLOPEVAR=variable | number

specifies the second disturbance variance parameter, $\sigma ^{2}_{2}$ , for trend types LL and DLL. Any nonnegative value, including 0, is permissible. If variable contains unknown parameters, they are estimated from the data. Similarly, if the SLOPEVAR= option is not specified, $\sigma ^{2}_{2}$ is estimated from the data.